Row space is orthogonal to nullspace

Let \(A\) be a matrix whose \(i^{th}\) row is represented as \(r_{i}^T\): \[ A = \begin{bmatrix} — & r_1^T & —\\ & \vdots & \\ — & r_n^T & — \end{bmatrix} \] Let \(x \in \mathcal{N}(A)\). Then \(Ax = 0\). Now: \[ \begin{aligned} Ax &= 0\\\\ \begin{bmatrix} — & r_1^T & —\\ & \vdots & \\ — & r_n^T & — \end{bmatrix} x &= 0\\\\ \begin{bmatrix} r_1^Tx\\ \vdots\\ r_n^Tx \end{bmatrix} &= 0 \end{aligned} \] It follows that \(r_i^Tx = 0\) for \(1 \leqslant i \leqslant n\). In other words, \(x\) is orthogonal to all the rows. Hence, we conclude that the nullspace is orthogonal to the row space.